279 
If the sphere is then to become only indeterminate, and 
not necessarily infinite, we must suppose that the numerator 
of the same expression (13) also vanishes; that is, we must 
have in this case the condition 
v.ABcDA = 0. (15) 
In words, as the product of the five successive sides of an 
uneven but rectilinear pentagon inscribed in a sphere, has 
been seen to be purely a line, so we now see that the product 
of the four successive sides of a quadrilateral inscribed in a 
circle is (in this system) purely a number: whereas, for every 
other rectilinear quadrilateral, whether plane or gauche, the 
grammarithm obtained as the product of four successive sides 
involves a grammic part, which does not vanish. This 
condition (15), for a quadrilateral inscribable in a circle, 
could not be always satisfied, when bp approached to a, 
and tended to coincide with it, unless the following theorem 
were also true, which can accordingly be otherwise proved : 
the product aBCA, or AB X BC X CA, Of three successive sides of 
any triangle ABC, ts a pure vector, in the direction of the tan- 
gent to the circumscribed circle, at the point a, where the 
sides which are assumed as first and third factors of the pro- 
duct meet each other. If, be the point upon this cireum- 
scribed circle which is diametrically opposite to a, we find for 
the length and direction of the diameter aa, in this notation, 
that is, for the straight line ¢v a from a,, the expression : * 
ABCA 
AA, = ; 
Vv . ABC 
(16) 
the denominator denoting a line which is in direction perpen- 
* With respect to the notation of division, in this theory, the author pro- 
poses to distinguish between the two symbols 
’ 
Q 
qa’ and rat 
